Browning, K. A.; Dillon, J. F.; Kibler, R. E.; McQuistan, M. T. APN polynomials and related codes. (English) Zbl 1269.94035 J. Comb. Inf. Syst. Sci. 34, No. 1-4, 135-159 (2009). Summary: A map \(f: \mathrm{GF}(2^m)\to \mathrm{GF}(2^m)\) is almost perfect nonlinear, abbreviated APN, if \(x\mapsto f(x+a)-f(x)\) is 2-to-1 for all nonzero \(a\) in \(\mathrm{GF}(2^m)\). If \(f(0) =0\), then this condition is euqivalent to the condition that the binary code of length \(2^m-1\) with parity-check matrix\[ H:=\left[ \begin{align*}{\ldots &\, \;\quad \omega^j\quad\ldots\cr \ldots& \, \, \;f(\omega^j)\, \;\ldots}\end{align*}\right] \]is double-error-correcting, where \(\omega\) is primitive in \(\mathrm{GF}(2^m)\).We give a brief review of these maps and their polynomials; and we present some new examples along with some related codes and designs which serve as invariants for their equivalence classes. Cited in 2 ReviewsCited in 49 Documents MSC: 94B05 Linear codes (general theory) 11T06 Polynomials over finite fields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) Keywords:almost perfect nonlinear map; binary code; parity-check matrix; double-error-correcting; designs PDFBibTeX XMLCite \textit{K. A. Browning} et al., J. Comb. Inf. Syst. Sci. 34, No. 1--4, 135--159 (2009; Zbl 1269.94035)